Finite Group Databasebeta

← #27183034

Subgroups — #27183034

Order 12928.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265873C211yesmaximal, characteristic, complemented
2#20265531C211yesmaximal, characteristic, complemented
3#20265515C211yesmaximal, characteristic, complemented
4#20265493C211yesmaximal, characteristic, complemented
5#20265484C211yesmaximal, characteristic, complemented
6#20265411C211yesmaximal, characteristic, complemented
7#20265408C211yesmaximal, characteristic, complemented
8#20265389C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic, complemented
10#20265316C211yesmaximal, characteristic, complemented
11C4×#13108061C211yesmaximal, characteristic, complemented
12D4×#11717579C211yesmaximal, characteristic, complemented
13#20265173C211yesmaximal, characteristic, complemented
14C2×#15692179C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108061C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19C22×#11717579C2211yescharacteristic
20C22×#11717579C2211yescharacteristic
21#15692250—21no—
22#15692247C2211yescharacteristic
23#15692237—21no—
24#15692229C2211yescharacteristic
25#15692229C2211yescharacteristic
26#15692227—21no—
27C4×D404C2211yescharacteristic
28#15692215—21no—
29#15692214C2211yescharacteristic
30C2×C4⋊Dic101C2211yescharacteristic
31#15692188C2211yescharacteristic
32#15692186C2211yescharacteristic
33#15692186C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35#15692179C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692179C2211yescharacteristic
39#15692179C2211yescharacteristic
40#15692179C2211yescharacteristic
41#15692176C2211yescharacteristic
42#15692176C2211yescharacteristic
43#15692176C2211yescharacteristic
44#15692175C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46C22⋊#11717579C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48C2×#41×C101C2211yescharacteristic
49C22×Q8×C101C2211yescharacteristic
50#124×C101—21no—
51#124×C101C2211yescharacteristic
52C22⋊Q8×C101—21no—
53C22⋊Q8×C101C2211yescharacteristic
54#121×C101—21no—
55#121×C101C2211yescharacteristic
56C4×Q8×C101C2211yescharacteristic
57C4×D4×C101—21no—
58C4×D4×C101C2211yescharacteristic
59C2×D404C2311yescharacteristic
60C2×C22⋊D101—21no—
61C2×C22⋊D101—21no—
62C22×Dic101—21no—
63C2×C4×D101—21no—
64#13108061—22no—
65#13108061—41no—
66#13108059—41no—
67C2×#11717579—41no—
68C2×#11717579—41no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C4⋊Dic101—21no—
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79C22⋊Dic101—21no—
80#13108051—21no—
81#13108051—21no—
82#13108051—21no—
83C4×Dic101—21no—
84#41×C101—22no—
85#41×C101—41no—
86C2×Q8×C101—21no—
87C2×Q8×C101—22no—
88C2×Q8×C101—41no—
89C2×Q8×C101C2311yescharacteristic
90C2×Q8×C101C2311yescharacteristic
91C2×D4×C101—21no—
92C2×D4×C101—21no—
93C2×D4×C101D411yescharacteristic
94C2×D4×C101C2311yescharacteristic
95C22×C4×C101—21no—
96C22×C4×C101—21no—
97C22×C4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C4⋊C4×C101—21no—
100C4⋊C4×C101—21no—
101C4⋊C4×C101D411yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C22⋊C4×C101D411yescharacteristic
108C22⋊C4×C101D411yescharacteristic
109C42×C101—21no—
110C42×C101C2311yescharacteristic
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—42no—
119Q8×C101#4112yes—
120D4×C101—22no—
121D4×C101—21no—
122D4×C101—21no—
123D4×C101—42no—
124D4×C101—42no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—42no—
133C2×C4×C101—42no—
134C2×C4×C101C2×D411yescharacteristic
135C2×C4×C101C2×D411yescharacteristic
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2411yescharacteristic
139C22⋊D101—42no—
140C22⋊D101—81no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—81no—
145D404—21no—
146D404—21no—
147C4×D101—81no—
148#11717579—21no—
149#11717579—21no—
150#11717579—21no—
151#11717579—21no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101#13512yes—
161C22×C101C22×D411yescharacteristic
162C2×D101—41no—
163C2×D101—41no—
164C2×D101—41no—
165C4×C101—21no—
166C4×C101—21no—
167C4×C101—22no—
168C4×C101—21no—
169C4×C101—21no—
170C4×C101—41no—
171C4×C101—41no—
172C4×C101C2×#4111yescharacteristic
173C4×C101H(5,𝔽2)11yescharacteristic
174Dic101—41no—
175Dic101—41no—
176Dic101—41no—
177C2×C101—22no—
178C2×C101—41no—
179C2×C101—81no—
180C2×C101#54611yescharacteristic
181C2×C101C2×#13511yescharacteristic
182C2×C101#57711yescharacteristic
183D101—81no—
184#3301—1011nomaximal, Sylow
185C101#330111yescharacteristic, complemented, Sylow
186#553—1011no—
187#548D10111yescharacteristic, complemented
188#488—1011no—
189#485—1011no—
190#481—1011no—
191#479—1011no—
192#466—1011no—
193#465—1011no—
194#463—1011no—
195#449—1011no—
196#448—1011no—
197C4×SD16—1011no—
198C8×D4—1011no—
199#416—1011no—
200C2×#103—1011no—
201C2×#41C2×D10111yescharacteristic
202C22×Q8C2×D10111yescharacteristic
203C2×SD16—1011no—
204C2×SD16—1011no—
205C2×D8—1011no—
206C22×C8—1011no—
207C22×C8—1011no—
208#126—2021no—
209#125—1011no—
210#124—21no—
211#124C2×D10111yescharacteristic
212#123—2021no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D10111yescharacteristic
215#121—21no—
216#121—1011no—
217#121—1011no—
218#121—2021no—
219#121C2×D10111yescharacteristic
220C4×Q8C2×D10111yescharacteristic
221C4×D4—21no—
222C4×D4—1011no—
223C4×D4—2021no—
224C4×D4C2×D10111yescharacteristic
225#117—1011no—
226C2×C4⋊C4—1011no—
227#107—1011no—
228#106—1011no—
229#106—1011no—
230C4⋊C8—1011no—
231#103—1011no—
232#103—1011no—
233#103—1011no—
234#103—1011no—
235#103—1011no—
236#103—1011no—
237#102—1011no—
238#102—1011no—
239#102—1011no—
240#102—1011no—
241C22⋊C8—1011no—
242C22⋊C8—1011no—
243C4×C8—1011no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D10111yescharacteristic
250C2×Q8C22×D10111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—1011no—
254C2×D4—2021no—
255C2×D4—2021no—
256C2×D4C22⋊D10111yescharacteristic
257C2×D4C22×D10111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2021no—
261C22×C4—2021no—
262C22×C4C22×D10111yescharacteristic
263C22×C4C22×D10111yescharacteristic
264SD16—4041no—
265SD16—2022no—
266D8—4041no—
267C2×C8—1011no—
268C2×C8—1011no—
269C2×C8—1011no—
270C2×C8—1011no—
271C2×C8—4041no—
272C2×C8—4041no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—1011no—
276C4⋊C4—1011no—
277C4⋊C4—1011no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4C22⋊D10111yescharacteristic
282C4⋊C4C22×D10111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2021no—
288C22⋊C4—2021no—
289C22⋊C4—2021no—
290C22⋊C4—2021no—
291C22⋊C4C22⋊D10111yescharacteristic
292C22⋊C4C22⋊D10111yescharacteristic
293C42—21no—
294C42—2021no—
295C42C22×D10111yescharacteristic
296C23—21no—
297C23—2021no—
298C23C2×C22⋊D10111yescharacteristic
299C23C2×C22⋊D10111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1310807312yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—8081no—
311D4—4042no—
312D4—2021no—
313D4—2021no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—8081no—
324C2×C4—8081no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4C2×C22⋊D10111yescharacteristic
329C2×C4C2×C22⋊D10111yescharacteristic
330C2×C4C2×C22⋊D10111yescharacteristic
331C2×C4C2×C22⋊D10111yescharacteristic
332C2×C4C23×D10111yescharacteristic
333C8—2021no—
334C8—2021no—
335C8—2021no—
336C8—2021no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—4041no—
346C22—4041no—
347C22—4041no—
348C22#1569227512yes—
349C22C22×C22⋊D10111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—4041no—
358C4—4041no—
359C4—4041no—
360C4C2×#1310807311yescharacteristic
361C4#1569230011yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—8081no—
366C2#2026582111yescharacteristic
367C2C2×#1569227511yescharacteristic
368C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#548×C101C2
socleC22×C101C22×D4
hypercenter#548D101
hypocenterC101#3301
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20265873C2semidirect
2. #20265531C2semidirect
3. #20265515C2semidirect
4. #20265493C2semidirect
5. #20265484C2semidirect
6. #20265411C2semidirect
7. #20265408C2semidirect
8. #20265389C2semidirect
9. #20265322C2semidirect
10. #20265316C2semidirect
11. C4×#13108061C2semidirect
12. D4×#11717579C2semidirect
13. #20265173C2semidirect
14. C2×#15692179C2semidirect
15. #548×C101C2semidirect
185. C101#3301semidirect
187. #548D101semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202658732
#202655312
#202655152
#202654932
#202654842
#202654112
#202654082
#202653892
#202653222
#202653162
C4×#131080612
D4×#117175792
#202651732
C2×#156921792
#548×C1012
#3301101